Modification of instability processes by multiplicative noises

We study two dynamical systems submitted to white and Gaussian random noise acting multiplicatively. The first system is an imperfect pitchfork bifurcation with a noisy departure from onset. The second system is a pitchfork bifurcation in which the noise acts multiplicatively on the non-linear term of lowest order. In both cases noise suppresses some solutions that exist in the deterministic regime. Besides, for the first system, the imperfectness of the bifurcation reduces the regime of on-off intermittency. For the second system, the unstable mode can achieve a jump of finite amplitude at instability but without hysteresis. We finally identify a generic property that is verified by the stationary probability density function of the dynamical variable when a control parameter is varied.     

Parameter dependence of stochastic resonance in the stochastic Hodgkin-Huxley neuron

Recently, the phenomena of stochastic resonance (SR) have attracted much attention in the studies of the excitable systems under inherent noise, in particular, nervous systems. We study SR in a stochastic Hodgkin-Huxley neuron under Ornstein-Uhlenbeck noise and periodic stimulus, focusing on the dependence of properties of SR on stimulus parameters. We find that the dependence of the critical forcing amplitude on the frequency of the periodic stimulus shows a bell-shaped structure with a minimum at the stimulus frequency, which is quite different from the monotonous dependence observed in the bistable system at a small frequency range. The frequency dependence of the critical forcing amplitude is explained in connection with the firing onset bifurcation curve of the Hodgkin-Huxley neuron in the deterministic situation. The optimal noise intensity for maximal amplification is also found to show a similar structure.     

Thermally assisted current-driven bistable precessional regimes in asymmetric spin valves

Spin-transfer torque in asymmetric spin valves can destabilize both parallel and antiparallel configurations and can lead to precessional modes also in the absence of an external magnetic field. We find a bistable precessional regime in such systems and show that thermal fluctuations can excite transitions (telegraph noise) between the corresponding oscillatory regimes that are well separated by irreversible paths at low temperatures. Because of the thermally induced transitions, the frequency of the resulting current-driven oscillations is different from that obtained at very low temperatures. We also show that the power spectrum in the bistable region is dominated by the out-of-plane oscillatory mode.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

单模光学双稳系统的不稳定区整体分布及分岔与混沌

本文根据半经典Maxwell-Bloch方程单模光学双稳系统的稳定性判据,在Δ-θ参数平面上描绘该系统不稳定区的整体分布(这里Δ和θ分别正比于原子与腔相对输入场的失谐度)。在已知的不稳定区中,具体讨论了系统的分岔与混沌性质。在双稳曲线的上、下支上同时发现混沌现象,找到了与稳定定态解共存的新的非定态解。 关键词：     

Stochastic resonance in underdamped,bistable systems

We carry out a detailed numerical investigation of stochastic resonance in underdamped systems in the nonperturbative regime. We point out that an important distinction between stochastic resonance in overdamped and underdamped systems lies in the lack of dependence of the amplitude of the noise-averaged trajectory on the noise strength, in the latter case. We provide qualitative explanations for the observed behaviour and show that signatures such as the initial decay and long-time oscillatory behaviour of the temporal correlation function and peaks in the noise and phase averaged power spectral density, clearly indicate the manifestation of resonant behaviour in noisy, underdamped bistable systems in the weak to moderate noise regime.     

Effect of phase noise on parametric instabilities

We report an experimental study on the effect of an external phase noise on the parametric amplification of surface waves. We observe that both the instability growth rate and the wave amplitude above the instability onset are decreased in the presence of noise. We show that all the results can be understood with a deterministic amplitude equation for the wave in which the effect of noise is just to change the forcing term. All the data for the growth rate (respectively the wave amplitude), obtained for different forcing amplitudes and different intensities of the noise, can be collapsed on a single curve using this renormalized forcing in the presence of noise.     

多频谐和与噪声作用下Flickering振子的安全盆侵蚀与混沌

研究了催化反应Flickering振子在多频率确定性谐和外力和有界随机噪声联合作用下,系统安全盆的侵蚀和混沌现象.将Melnikov方法推广到包含有限多个频率外力和随机噪声联合作用的情形,推导出了系统的随机Melnikov过程,根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值,然后用数值模拟方法计算了系统的安全盆分岔点.结果表明,由于随机扰动的影响,系统的安全盆分岔点发生了偏移,并且使得混沌容易发生.同时证明,激励频率数目的增加扩大了参数空间上的混沌区域,也使得安全盆分岔提 关键词： 多频率激励 Flickering振子 安全盆 混沌     

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ～τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t?→?2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.     

Limit cycles of the ballast resistor caused by intrinsic instabilities

Spatio-temporal patterns of the ballast resistor are investigated. It is well known that in a voltage-controlled ballast resistor an electrothermal instability leads to stable stationary states consisting of hot and cold domains. Such states may become oscillatory unstable, giving rise to the bifurcation of limit cycles. These limit cycles are not caused by the external circuit but by a recently proposed novel intrinsic mechanism. There are two types of oscillatory instabilities: bulk instabilities and boundary-induced instabilities. The bulk instabilities are caused by resistivities which are not monotonically increasing functions of the temperature. The boundary-induced instabilities occur in small systems with Neumann boundary conditions. To find the bulk instability, experiments with materials showing a metal-semiconductor transition or high-temperature superconductors are suggested. To understand these new phenomena, the equation of motion is reduced to ordinary differential equations where the instabilities can be discussed analytically.     

Spectrum of acoustic radiation caused by cavitation: Analytical model

Nonlinear dynamics of bubbles in liquid in the presence of the resonance and noise acoustic fields is analyzed. The effect of fluctuations associated with the random field component is found to be most pronounced in the vicinity of the bifurcation values of the field amplitude and detuning, these values corresponding to changes in the number of stable oscillatory states of a bubble. The radiation spectrum of a single bubble in the vicinity of the fundamental resonance is determined. In the framework of the proposed model, a comparison of this spectrum with the real spectrum of acoustic radiation caused by cavitation is performed.     

Synchronization of noisy systems by stochastic signals

We study, in terms of synchronization, the nonlinear response of noisy bistable systems to a stochastic external signal, represented by Markovian dichotomic noise. We propose a general kinetic model which allows us to conduct a full analytical study of the nonlinear response, including the calculation of cross-correlation measures, the mean switching frequency, and synchronization regions. Theoretical results are compared with numerical simulations of a noisy overdamped bistable oscillator. We show that dichotomic noise can instantaneously synchronize the switching process of the system. We also show that synchronization is most pronounced at an optimal noise level-this effect connects this phenomenon with aperiodic stochastic resonance. Similar synchronization effects are observed for a stochastic neuron model stimulated by a stochastic spike train.     

Frequency and phase synchronization in stochastic systems

The phenomenon of frequency and phase synchronization in stochastic systems requires a revision of concepts originally phrased in the context of purely deterministic systems. Various definitions of an instantaneous phase are presented and compared with each other with special attention paid to their robustness with respect to noise. We review the results of an analytic approach describing noise-induced phase synchronization in a thermal two-state system. In this context exact expressions for the mean frequency and the phase diffusivity are obtained that together determine the average length of locking episodes. A recently proposed method to quantify frequency synchronization in noisy potential systems is presented and exemplified by applying it to the periodically driven noisy harmonic oscillator. Since this method is based on a threshold crossing rate pioneered by Rice the related phase velocity is termed the Rice frequency. Finally, we discuss the relation between the phenomenon of stochastic resonance and noise-enhanced phase coherence by applying the developed concepts to the periodically driven bistable Kramers oscillator.     

Eckhaus instability in systems with large delay

The dynamical behavior of various physical and biological systems under the influence of delayed feedback or coupling can be modeled by including terms with delayed arguments in the equations of motion. In particular, the case of long delay may lead to complicated and high-dimensional dynamics. We investigate the effects of delay in systems that display an oscillatory instability (Hopf bifurcation) in the absence of delay. We show by analytical and numerical methods that the dynamical scenario includes the coexistence of multiple stable periodic solutions and can be described in terms of the Eckhaus instability, which is well known in the context of spatially extended systems.     

Noise-induced coupling and phase transition in initially homogeneous bistable system

We show that a colored spatial noise induces a heterogeneous behavior and coupling of initially uncoupled single bistable units. A formal approximation reduces a non-Markovian stochastic process described by the initial set of equations into Markovian process in terms of Langevin equation, for which a simple piecewise linear emulation was used to represent the nonlinear deterministic force. It turned out that the coupling leads to a phase transition due to the noise-induced diffusive term. As an example, a typical bistable noisy system with symmetric double-well potential was studied.     

Harmonic noise: Effect on bistable systems

The coordinate of a white noise driven harmonic oscillator is used as a stochastic source term in bistable dynamics. This new kind of Gaussian colored noise gives rise to resonance phenomena due to a peak in the spectrum. We investigate its effect on linear and bistable systems. We derive a Markovian approximation for driven bistable oscillators and overdamped systems. In the resonance region computer simulations were carried out using an extension of Fox' algorithm procedure for colored noise. We find an increase of the transition rates in bistable systems as compared with the case of bistable systems driven by white and exponentially correlated noise.     

Controlling bistability in microelectromechanical resonators

For a microelectromechanical (MEM) resonator, the combination of mechanical nonlinearity and electrical driving force can lead to bistability. In such a case, the system exhibits two coexisting stable oscillatory states (attractors): one with low and another with high energy. Under the influence of noise, with high probability the system can be perturbed into the low-energy state. We propose a robust control scheme to place the system in the high-energy state. Our idea is not to pull the system out of the bistable regime but instead to take advantage of the nonlinear dynamics to achieve high-energy output. In particular, our control scheme consists of two steps: bifurcation control that temporarily drives the system to a regime with only one attractor, one that is the continuation of the high-energy attractor in the bistable regime; and ramping parameter control that restores the bistability while maintaining the system in the high-energy attractor. We derive an analytic theory to guide the control, provide numerical examples, and suggest a practical method to realize the control experimentally. Our result may find potential usage in devices based on MEM resonators where high output energy is desired.     

光学双稳态和混沌运动中的临界现象

本文以具有反馈时间延迟的非线性系统弛豫方程为出发点,讨论了双稳态临界点的临界现象。发现双稳区边沿的临界慢化与混沌运动中切分叉点附近阵发混沌时间间隔发散具有一致性,临界指数为1/2。在双稳区(尖顶突变模型)的尖顶处,与双稳区边沿临界点不同,它与混沌运动中倍周期分叉点及劈分叉点的临界慢化具有一致性,临界指数为1。上述结果具有普适性。以液晶混合光学双稳系统为例,进行了计算机实验,其结果与分析相一致。 关键词：     

Inversion in harmonic noise driven bistable oscillators

We consider the dynamics in a asymmetrical bistable oscillator driven externally by a noisy harmonic oscillator. The basic for our approach is the energy dynamics in each well of the oscillator in the low friction limit. We discuss the possibility of energy inversion states generated by harmonic noise and show this effect by simulations.